Question: $\vec u = 6\hat i +(-8)\hat j$ Find the direction angle of $\vec u$. Enter your answer as an angle in degrees between $0 ^\circ$ and $360^\circ$ rounded to the nearest hundredth. $\theta =$
Answer: What is a direction angle? The direction angle, $\theta$, of $\vec{u}$ is the angle between the positive $x$ -axis and $\vec{u}$. $y$ $x$ $(6, -8)$ $\vec u$ ${\theta}$ Using the inverse tangent function Let's think about the components of $\vec u$ and use the inverse tangent function, $\tan^{-1}$ (also sometimes called arctangent and written as $\arctan$ or $\text{atan}$ ) to find $\theta$. $y$ $x$ $(6, -8)$ $\vec u$ ${\theta}$ $-8}$ $6}$ $\theta = \text{tan}^{-1} \left ( \dfrac{\text{Vertical component}}{\text{Horizontal component}} \right) ~~~$ $\theta=\text{tan}^{-1}\left(\dfrac{-8}{6}\right)$ $\theta\approx{-53.13^\circ} {~~~~~~~\text{WARNING: This is not the correct answer.}}$ This makes sense because ${-53.13^\circ}$ is in the fourth quadrant, and $\vec u$ is in the fourth quadrant, BUT the question asked us for an angle between $0^\circ$ and $360^\circ$. Adding $360^\circ$ because $\vec u$ is in the fourth quadrant Key idea: The inverse tangent function only outputs values between $-90^\circ$ and $90^\circ$, which is why our calculator didn't give us the answer we were looking for. We need to add $360^\circ$ (which doesn't change the location of the angle!) to get an angle between $0^\circ$ and $360^\circ$. $y$ $x$ $(6, -8)$ $\vec u$ ${306.87^\circ}$ $~~~~~~~~-53.13^\circ$ $\theta \approx {-53.13^\circ} + 360^\circ$ $\phantom{\theta} \approx 306.87^\circ$ The answer $\theta \approx 306.87^\circ$